lnstructions: Answer all four [4] - EPrints USM

UNIVERSITI SAINS MALAYSIA Second Semester Examination

Academic Session

200512006

April/May 2006

MAT

161E

- Elementary Statistics [Statistik Permulaan]

Duration :3 hours [Masa

^{: 3}

jam]

Please check that this examination

^paper

consists

of

THIRTEEN pages of

^printed

material before you begin the

examination.

[Sila

^pastikan

bahawa kertas

peperiksaan

ini mengandungiT|GA

BELAS muka surat

yang bercetak sebelum anda

memulakan

peperiksaan inil.

lnstructions: Answer all four

_[4]

questions.

Aranan:. Jawab semua empat [4] soalanl.

67

...12-

IMAT 161EI

1.

(a)

The

following

stem-and-leaf diagram gives the distances

(in

thousands

of

mites) driven during the past year by a sample of drivers

in

a

city

^.

2

0

I

2

aJ 4 5 6

369 285 516

8

I

105

Key:

2

tl z = 12

thousand miles

(i)

Compute

the

sample mean, median, and mode

for

the data

on

distances driven.

(ii)

Compute

the

standard deviation and the interquartile range for these data.

(iii)

Make a box-and-whisker

plot for

the data and describe

the

shape

of

the distribution.

(iv)

^Between

the

standard

deviation and the interquartile

range,

which

is preferable as a measure of the variation of the above data? Why?

O) A

^{thief has} stolen an automatic teller machine

(ATM)

card. The card has a four-

digit

personal

identification

nurnber

(PnD. The thief knows that the first

two

digits

are

2

and 6, but he does not

know

the last rwo digits. Thus, the

PIN

could be any number from 2600 to 2699.

(i)

How many possible numbers can the thief attempt for the last two digits?

(iD

The automatic teller machine

will

not allow more than three unsuccessful attempts

to

enter the

PIN. After

the

third wrong PlN,

the machine keeps the card and allows no further attempts. Find the

probability

that the

thief

finds the conect PIN

within

the three attempts? (Assume that the thief

will

not try the wrong PIN twice.)

(c) A

random sample

of

80

lawyers

was selected, and they were asked

if

they are

in

favour

of

or against the death penalty. The

following

table shows the distribution of their responses by age.

Favours Neutral

l5

15 10 5

10 15

5

2

<

3s (v)

3s -

4s

^(M)

Age (yearc)

68

...13-

IMAT 161EI

(i) If

^a lawyer is randomly selected from this sample, what

is

the probability that he/she opposes or is neutral about the death penalty?

(ii)

Two lawyers are selected at random from those

in

favour the death penalty.

Calculate the probability that one is less than 35 years old and the other is more than 35 years old.

Given that90%o of the lawyers in favour of the death penalty are males, as do 70o/o

of

^{those who} oppose and3}o/o of those who are neutral.

(iii)

What is the probability that a lawyer selected ^at random is ^a male?

(iv)

Suppose

that a lawyer

selected

at random is a male. What is

^the

probability that he is in favour of the death penalty?

[00

^marks]

I.(a)

Gambarajah tangkai dan daun yang berilatt menunjukkan

jarak

(dalam ribuan batu) yang dipandu oleh suatu ^sampel pemandu di sebuah bandaraya.

105

Petunjuk: I /2 : 12 ribubatu

Hitung

min sampel, median sampel dan mod sampel bagi data mengenai

jarakyang

dipandu.

Hitung sisihan

piawai

dan

julat

antara

lanrtil

bagi data di atas

Buat suatu

plot

kotak bagi data di atas dan perihalknn bentuk taburannya.

Antara sisihan

piawai

dan

julat

_antara

lanrtil,

sukatan apakah yang lebih dicenderungi sebagai sukatan seralwn bagi data di atas? Kenapa?

(b) _Seorang

pencuri

telah mencuri sekeping

lad

mesin

pelayan

automatik

(AfM.

Kad tersebut

mempunyai

nombor

pengenalan

peribadi (PIN) yang terdiri daripada

empat

digit.

Pencuri tersebut tahu bahawa dua

digit

pertama

ialah

2

dan 6, tetapi ia tidak

tahu dua digit

yang terakhir.

Oleh

itu, PIN

l(ad tersebut mungkin sebarang nombor daripada 2600 sehingga 2699.

(,

Berapalcah nombor yang mungkin dicuba oleh pencuri tersebut untuk dua digit yang terakhir?

369 285 516

8

I

2 0

I

2 3 4 5 6

(, (i, (ii, (tt

...14-

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5 IMAT 161E]

2.(a)

The discrete random variableXhas a probability function given

by

P(X=x)=

1<x<5 x=6

otherwise

(i)

Determine the value

of

C, ^a constant.

(ii) Find

^P

(x ssl x, +).

(iii)

Find E(,Y) and show

thatYar(X):

1.6553

(iv) Approximate the probability that a sample of 100

independent observations

ofXhas

^a total sum of less than 200.

O) A history

^{teacher gives}

a

test consisting

of

50 multiple-choice questions. Each question has

four

choices

of which only

one

is

correct.. The scoring includes ^a penalty

for

guessing; each correct answer

is worth I point,

^each

wrong

^answer

costs 1/2 point and an unanswered question gets 0

point.

For example,

if

a student answers

35

questions correctly,

8

questions

incorrectly,

and does

not

answer ⁷ questions, the total score for the student

will

be 35(1)

+ 8(-l/2)

⁺ 7(0)

:

31.

(i)

Suppose a student has no idea what the correct answer to a question is.

By

guessing

the

answer, does

he

increase

or

decrease

his

expected score?

Explain.

Suppose

a

student answers

38

questions correctly and ^guesses

on

the

other

12 questions by randomly choosing one of the four answers for each.

(ii)

What is the probability that the student

will

score more than 40 points?

(iii)

What is his expected score?

(c)

The number

of

cars

arriving

at a

petrol

station

in

a period

of r

minutes may be assumed to have a Poisson distribution with mean

Z.

20

What

is

the

probability

that fewer than

6

cars

will

arrive

in a

10 minute period?

Suppose a car has

just

arrived at the petrol station. What is the probability that

it will

be at least 5 minutes before the next car arrives?

Approximate the probability that more than

24

cars

will

arrive in an hour.

[100 marks]

?1

(r)' ttj

C 0

(D (iD (iiD

.../6-

IMAT 161EI

2.(a)

Pembolehubah

rawak

dislcrit

X

mempunyai

fungsi

kebarangknlian seperti yang berilan:

P(x ='={

[r)' \2)

C 0

l<-r<5 x=6

di

tempat

lain

(,

Tentulrnn

nilai

C, suatu pemalar.

(i,

^Dapatkan

P(x <sl x >

^+).

(ii, Dapatkan

E(X) dan tunjul<kan bahawa Var(X)

=

^1.6553

(iv) Dapatkan

kebarangkalian

hampiran

bahawa

suatu

sampel

rawak

100 cerapan tak bersandar

X

^mempunyai

jumlah

kurang daripada 200.

(b) Seorang

guru sejarah

memberi

suatu ujian yang terdiri daripada 50

soalan

pilihan

berganda. Setiap soalan mempunyai empat

pilihan jawapan yang

mana hanya satu sahaja adalah betul, Pemarkahan

ujian

termasuk suatu

penalti

untuk penelraan; setiap

jawapan

betul

bernilai I marlah,

setiap

jawapan

_salah

ditolak

%

marlah

dan soalan yang

tidak

dijawab mendapat 0 marknh. Contohnya, jilca seorang

pelajar

menjawab 35 soalan dengan betul, 8 soalan dijawab salah dan 7

soalan tidak dijawab, maka

jumlah

markah yang akan

diperolehnya ialah 35(l)

+

8(-I/2) + 7(0):31.

O

^{Andaikan seorang}

pelajar tidak

^{tahu langsung jawapan}

yang

betul bagi suatu soalan. Dengan ^meneka jawapannya, _adaknh

ia

meningkatknn atau menurunkan markah yang dijangkn? Jelaskan jawapan _anda.

Andailcan seorang

pelajar

menjawab 38 soalan dengan betul dan menekn pada 12 soalan

yang lain

dengan

memilih

secara

rawak satu daripada

empat

pilihan

jawapan bagi setiap satu.

(i, Apakah

kebarangkalian bahawa

pelajar

tersebut

aknn

mendapat lebih daripada 40 marknh?

(ii,

Berapakah markahyang

dijanglu

diperolehnya?

(c)

Bilangan kereta

yang tiba di

sebuah stesen minyak

dalm

suatu tempoh

t

minit, dapat diandaiknn tertabur secara

Pa'

t$son aengan

'

mtn

' _. lt

(i)

Apaknh kebarangknlian bahawa kurang daripada 6 buah lcereta alean tiba dalam sudtu tempoh

I0

minit?

72

...17-

IMAT 161EI

(i, Andaiknn

sebuah

kereta baru sahaja tiba di

stesen

minyak

tersebut.

Apaknh kebarangkalian bahawa selatrang-kurangnya 5 minit akan berlalu sebelum kereta seterusnya tiba?

(ii, Dapatknn

kebarangkalian

hampiran bahawa lebih daripada 24

buah kereta akan tiba dalam satu jam.

fl00

^markahJ

3.(a) A

bag contains a very large number of marbles, identical except

for

their colour.

Of

these, an unknown

proportionp

are blue. Suppose you are required to test the

null

hypothesis .F/o i

p

₌ 0.2 against the alternative

H,

^:

p

<

0.2. To

perform the test, you are to take a random sample

of

100 marbles and note

X,

the

number of

blue marbles in the sample.

(i) If the

significance

level is

10%, determine whether the

null

hypothesis should be accepted

when

X

=

15.

(ii) If the

significance

level is to be

as close as possible

to

^5o/o,

find

the rejection region in the form

of

0 <

X ( c,

where a is an integer.

(iii)

Calculate the power

of

the test when the rejection region

is

0 <

X <I4,

and _{P = 0'1}

(b)

A drug

manufacturer produces

250-milligram

capsules

of a new antibiotic. A random

sample

is

selected,

ffid the

amount

of antibiotic in

each capsule is determined. The results are as follows (in milligrams):

7

(D

252, 246, 242, 250, 255, 258, 250, 252, 250,

258.

\x=2,513 Z*' =631,74I

Find

a 95% confrdence

interval

for

p,

the mean amount

of

antibiotic per capsule.

Determine the

minimum

number

of

capsules

that

^you need

to

sample

if

you

would like

to be 99%o confident that an estimate

of

the mean

arlount

of antibiotic per capsule is

within I

milligram of the true mean.

Over a period of months, the manufacturer found that

of

a sample

of

150 capsules,

18

capsules

have

unacceptable

amounts of the

antibiotic.

Calculate an approximate 95o/o confidence

interval for

the proportion

of

capsules having an acceptable amount of the antibiotic.

(ii)

(iiD

(c) Attitude toward mathematics was measured for

two

different groups. The attitude scores range

from 0 to 80 with the higher

scores

indicating a more

positive attitude.

One group

consisted

of

Elementary Education majors,

ffid the

other

goup

consisted of majors from several other areas. The data are shown below:

73

...t8-

8

IMAT 161EI

Elementary Education

(l)

Non-Elementary Education (2)

75 110

42.7 49.3

15.5

t7.0

(i)

Assuming a coflrmon population variance, obtain a pooled estimate

of

this variance.

(ii)

Construct

a

95o/o confidence

interval for

p,

- /rz,

^the difference between

the mean attitude

scores

of

Elementary Education

majors and

Non- Elementary Education maj ors.

(iii) Using the p-value

method, test that Non-Elementary Education majors have

a more positive

attitude toward mathematics

than the

Elementary Education majors ^at the the 5% significance level.

[100 marks]

3.(a)

Sebuah beg mengandungi suatu bilangan besar

guli

yang serupa tetapi berbeza warna.

Daripada

bilangan

ini,

suatu kndaran

p

yang tidak diketahui adalah

guli

warna biru. Andaikan anda

perlu

menguji hipotesis

nol

H _{o :}

p

₌

0.2

berlawanan hipotesis

alternatif H,

^:

p

< 0.2.

Bagi

menjalankan ujian, anda

perlu

mengambil suatu sampel

rawak

100

biji guli

dan mencatit

nilai X, iaitu

bilangan

guli biru

dalam sampel tersebut.

Jilra aras keertian

ujian ialah

10%o, tentuknn sama ada hipotesis nol

patut diterimaapabilaX=

15.

Jika aras keertian ujian

dikehendaki sedekat

mungkin dengan

594,

dapatknn kawasan penolakan dalam

bentuk

0 <

X <a, a ialah

suatu integer.

Hitung kuasa ujian apabila

kawasan

penolalmn ialah 0<X <14,

dan

P=0.1

(b) Sebuah

syarikat pengeluar ubat

menghasilkan sejenis

antibiotik baru

dalam

bentuk knpsul-kapsul

250

milligram.

Suatu sampel rawak

dipilih dan

amaun

antibiotik dalam

setiap kapsul ditentukan.

Hasilnya ialah

seperti

yang

berikut (dalam

milligram):

252, 246, 242, 250, 255, 258, 250, 252, 250,

258.

lx

⁼

2,513 Zr' =

^631,741

(, Dapatkan suatu selang keyakinan 95% bagi p , iaitu min

^amaun

antibiotik di dalam setiap kapsul yang dihasilkan.

(,

(it)

74

.../9-

IMAT 161EI

(i,

Tentukan

bilangan

kapsul

yang

minimum yang

perlu

disampelkan

jilw

anda ingin 99% yakin bahawa suatu anggaran bagi min amaun

antibiotik di

dalam setiap lcapsul adalah dalam sekitar

I milligram

^{daripada min}

yang sebenar.

(iit)

Dalam suatu tempoh beberapa bulan, pengeluar ubat tersebut mendapati bahawa daripada suatu sampel 150

biji

lcapsul, 18

biji

mempunyai amaun

antibiotik yang tidak

dapat diterima. Dapatkan suatu selang lcEtakinan 95% hampiran

bagi

lmdaran lcnpsul

yang

mempunyai amaun

antibiotik yang

dapat diterima.

G) Stkap terhadap Matematik diulatr bagi dua latmpulan pelajar yang berbeza.

Julat

bagi slcor sikap adalah daripada 0 sehingga 80 dengan skor

tinggi

menandakan

sikap yang lebih positif. Satu htmpulan terdiri daripada pelajar

dalam penglrhususan Pendidiknn Asas

dan

latmpulan

yang satu lagi terdiri

daripada

pelajar

dalam pengkhususan beberapa

disiplin lain.

Data

yang

diperoleh ialah seperti yang berikut:

Pendidikan Asas

(l)

Bulcan Pendidikan Asas (2)

75

II0

^42.749.3

t 5.5 17.0

Dengan

andaian

bahawa

varians

kedua-dua

populasi adalah

sepunya, dapatlrnn anggaran tergembeleng bagi varians tersebut.

Bina

suatu selang keyakinan

95% bagi l\- pz, iaitu

perbezaan antara min

bagi

skor siknp pelajar pengkhususan Pendidiknn Asas dan min bagi skor

pelajar

bukan pengkhususan Pendidikan Asas.

Dengan

menggunakan

kaedah nilai-p, uji bahawa pelajar

^bulcan

pengkhususan

Pendidikan Asas

mempunyai

silap yang lebih positif terhadap Matematik daripada pelajar

pengkhususan Pendidilcan Asas pada aras keertian 5%o.

fl00

^markahJ

4. (a) (D Explain why

nonempty,

mutually

exclusive events

A

^and

B must

be dependent.

(iD

Show

that if A

and

B

are independent events, then

A

and

B

^{are also}

independent.

(b) In

a2003 survey conducted in a city, respondents answered the question "What is

the

combined income

of your

household?"

The

responses appear

in the

table below

(r)

(i, (ii,

75

.../10.

IMAT 161EI

Household Income %o of Households

Under RM20,000

RM20,000 to less than RM40,000 RM40,000 to less than RM60,000 RM60.000 and over

25 43 22 10

Assume that the above results hold true for the entire population

in

the

city. In

^a

recent

poll of

200 households

in

the same

city,

39 had a

total

income

of

under

RM20,000, 83

had

an

income between

RM20,000

and

RM40,000, 45

had an income between RM40,000

to

less than RM60,000 and the rest had an income

over RM60,000. Test at the

5o/o

level of significance whether the

current percentages of the city's households in each income category are the same now as they were in 2003.

(c)

The

following

table lists the ages

(in

years

)

and the prices

(in

thousands

of RM)

for a sample of cars of the same model.

Age ^(x years)

Prt". (RM y

000)

8

40

11

30 4 95

2 130

6 80

)

90

13 15

lx

⁼

49 ly

⁼

a80 lxz = 435 Zy' = 43,150 Zry

^{= 2,415}

(i)

Find the least square regression line.

(ii)

Explain the meaning of they-intercept and the slope of the regression line.

(iii)

Calculate

r

and rz and explain theirmeaning.

(iv)

Test at

a =.0twhether p

is negative.

[100 marks]

(a) (l

Terangl<an kenapa

peristiwa A

dan

peristiwa B yang tak

kosong dan

s aling el<s Hus if s emes tinya ^s aling b ers andaran.

(i, Tunjukkan

bahawa

jika A dan B adalah peristiwa-peristiwa

saling bersandaran, makn

A

dan E

iuga

^{adalah saling} bersandaran.

(b)

Dalam

suatu tinjauan

pada

tahun 2003

yang

dijalanlcan

di

sebuah bandaraya, responden menjawab

soalan

"Berapakah

jumlah pendapatan

keluarga anda? ".

Jawapanyang diberi

ditunjul*an

dalam jadual yang berikut:

Pendapatan keluarga ^o% keluarga

Kurang daripada RM2 0,000

RM20,000 hingga kurang daripada kM40,000 RM40,000 hingga latrang daripada RM60,000 RM60,000 dan lebih

25 43 22

I0

10

76

.../,11-

IMAT 161E]

Andaikan hasil di atas adalah benar bagi seluruh populasi

di

bandaraya tersebut.

Dalam

suatu

tinjauan terkini

terhadap

200 buah

keluarga

di

bandaraya yang

sama, 39 buah keluarga

mempunyai

jumlah pendapatan larang

^daripada

RM20,000, 83 buah keluarga mempunyai

jumlah

pendapatan antara RM20,000

dan RM40,000, 45 buah keluarga

mempunyai

jumlah pendapatan

antara RM40,000 dan RM60,000 dan

yang

selebihnya mempunyai

jumlah

pendapatan

lebih daripada RM60,000. Uji pada aras keertian

50%

sama ada

peratusan semasa keluarga

di

bandaraya tersebut dalam setiap kntegori pendapatan adalah sama seperti dalam tahun 2003.

(c)

Jadual yang

berilal

menyenarailcan usia (dalam tahun) dan harga (dalam ribuan RM) bagi suatu sampel kereta yang sama model.

Usia (x tahun)

H"rga(RIttyW

8 40

II

30 4 95

2 130

6 80

5 90

13 15

\x

⁼

49 ly = 480 lxz

⁼

435 Zy'

⁼

43,150 Zry

^{= 2,415}

(,

Dapatlmn garis regresi lansa dua terkecil.

(i,

Terangknn malvta pintasan-y dan kecerunan garis regresi yang diperoleh.

(iiil

Hitung

nilai r

dan

nilai r'

dan teranglcan malcna

nilai

masing-masing.

(iu)

Uji

pada a =.01

sama ada

p

adalah negatif.

[100 ^marlcahJ

11

77

...112-

12 [MAT

^161EJ

FORMT]LA

i=-

S-=

Zx

n

Z*'

sr'= (n., -l)s.2

⁺

(n, -l)s r' nrln,-2

E.')'

n

X+Y

nx+ny

n-l

Confidence Interval

X

v

L-! La/2

o

--- in

(X -Y ) ! zo,r^lL+u G.,

^Y

\,. n,

+,

,s

-"d/z

!n

l-

(X -7 ) tto,,

trXrotz b,(r- i,)

Test Statistics

z X-p

o t-

an

X-p

/,= (f,,- f,r)-(n,- pr)

T=r p,(1-

p.

)

_{_}

pr(l- p, )

T

6 -h-@, - ar)

lor'

It)

, ov-

rl- ^T

\n*

-^nY

6 -n-Q', - t'r)

s

r

ln

7-pa

z- z- (b,-Py)-(n,-nr)

f-

-,( t /) p\t-p)l-+-l

\'"

^nY

)

G

s,.r ^T_

,-=Lt, , s! b-n)' E=nP

-_b-F,

.tb

.r.( t t)

Dp - ln- l-+-

^n,, _t/ ^l|

?9

...113-

13

IMAT 161EI

Regression and Correlation

Sxy

=Z*y - (I')(Iv)

sr= -b F-

C =-_1{ ^s"

".W

-oooOooo-

80